Fig. 18. 



Let the points P p be connected by a link, as before. Let a link P' x*, equal 

 in length to the link P />, be attached to the point P', and let the extremity x 1 

 of this link be connected with the point p by another link, equal in length to 

 P P', by pivots at x 1 and p, so that the figure P P' x' p shall be a jointed par- 

 allelogram, the angles of which will be capable of altering their magnitude 

 with every change of position of the rods o p and O P. Thus, when the rod 

 O P descends, the angles of the parallelogram at P and x' will be diminished 

 in magnitude, while the angles at P' and p will be increased in magnitude. 

 Now, let a line be conceived to be drawn from O to x'. It is evident that 

 that line will pass through the middle point of the link p P, for the triangle 

 O P a- is in all respects similar to the greater triangle O P' a/ only on half the 

 scale, so that every side of the one is half the corresponding side of the other. 

 Therefore P a: is half the length of P' a/ ; but P' x' was made equal to P p, 

 and therefore p x is half of P p ; that is to say, x is the middle point of P p. 



It has been already shown, that in the alternate motion of the rods o p, O P, 

 in ascending and descending, the point x is moved upward and downward in 

 a true vertical line. Now since the triangle O P a is in all respects similar 

 to P' x', and subject to a similar motion during the ascent and descent of 

 the rods, it is apparent that the point a/ must be subject to a motion in all re- 

 spects similar to that which affects the points x, except that the point x 1 will 

 move through double the space. In fact, the principle of the mechanism is 

 precisely similar to that of the common pantograph, where two rods are so 

 connected as that the motion of the one governs the motion of the other, so 

 that whatever line or figure may be described by one, a similar line or figure 

 must be described by the other. Since, then, the point x is moved upward 

 and downward in a vertical straight line, the point a/ will also be moved in a 

 vertical straight line of double the length. 



If such an arrangement of mechanism as has been here described can be 

 connected with the beam of the steam-engine, so that while the point x' is at- 

 tached to the top of the steam-piston, and the space through which it ascends 

 and descends shall be equal to the length of the stroke of that piston, the point 

 x shall be attached to the rod of the air-pump piston, the stroke of the latter 

 being half that of the steam-piston, then the points a:' and x will guide the 

 motion of the two pistons so as to preserve them in true vertical straight lines. 



The manner in which these ideas are reduced to practice admits of easy ex- 



